Tagged Questions

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Use complex number to solve this equation $\int e ^{3x} cos x dx$?

I can solve it another way, but am not sure how to use complex numbers to solve it. Thanks for your help
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A problem on orthonormality of a set of complex functions

The following is a problem of an undergraduate exam test:
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Integration Error

Sorry if this doesn't make any sense or if I did something obviously wrong, I was just playing around with taylor series' and then I got stuck. I know from the geometric series that: ...
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An Integration Calculation

I'm just having a bit of difficulty understanding the last couple of steps made in the paper Horowitz & Hubeny - Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium (p.8) ...
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Why is $\int |e^{ix}|^2 dx = x + C$?

Quick question: Wolfram Alpha tells me that $$\int |e^{ix}|^2 dx = x + C$$ Why is that?
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Complex integral and parametrization of a circle

I am trying to compute the following integral of $$\int \frac{1}{z^3+3} dz$$ over a circle of radius $2$, centerd at $(2,0)$. Thus I am trying to compute the residue and have found that the function ...
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Calculating a complex integral by rewriting as a contour integral on |z|=1.

I need to show that $\int_0^{2\pi}\frac{d\theta}{2+i\:sin\theta}=\frac{2\pi}{\sqrt{5}}$ I used $sin\theta=\frac{1}{2i}(e^{i\theta}-e^{-i\theta})$ and substituted $z=e^{i\theta}$. I ended up with ...
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Complex integration around a branch point

I am confused about the "deformation" of a closed contour that my book is doing. For reference, it is example 2.4.3 on pg. 75-76 from this free online book. The example is the integration of 1/z ...
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Integral of complex questions?

$$\int_0^{\pi/4} \frac {\sin x + \cos x}{\sin^4x+\cos^2x}dx$$ $$\int e^x\cot x(\csc x-1)dx$$ These two integrals are impossible to find. If anyone knows how to integrate them please help me. I am ...
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How can I integrate this zeta function expression?

Can you integrate this function: $$f(k)=\exp\left(-\Re\left(\sum\limits_{n=1}^{n=scale} \frac{1}{n} \zeta(1/2+i \cdot k)\sum\limits_{d|n} \frac{\mu(d)}{d^{(1/2+i \cdot k-1)}}\right)\right)$$ with ...
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Why isn't $\int\sin(ix)~dx$ equal to $i\cos(ix)+C$ ?

I was playing around with imaginary numbers, and I tried to solve $$\int\sin(ix)~dx$$ and ended up getting $$i\cos(ix)+C$$ But apparently the answer is $$i\cosh(x)+C$$ I was just wondering, is this ...
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Integration $1/x$ - complex number

Why there is no integral $$\int_{-e}^{e}\frac{1}{x}$$ And why integral $$\int_{-e}^{-1}\frac{1}{x}= -1$$ and not $$\int_{-e}^{-1}\frac{1}{x}=(-1 + i\cdot\pi)$$ E.g. ...
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Integral of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ The function $$F(z)=\sum_{n=1}^\infty a_nz^n$$ converges in $|z|<1$. How can I ...
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Integral over a complex plane

I am wondering if closed-form solutions exist for the following integral: $$\int_\mathbb{C}e^{-\frac{|u-v|^2}{c}-|v|^2}|v|^{2n}d^2v$$ where $u$ is a complex number, $c>1$ a real constant and $n$ ...
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Zeroes of $s+\sum\limits_{n=2}^\infty \frac{(-1)^{n+1}}{n^s\ln n}$?

Where are the solutions of the equations $$s+\sum\limits_{n=2}^\infty \dfrac{1}{n^s\ln n}=0\quad \text{and}\quad s+\sum\limits_{n=2}^\infty \dfrac{(-1)^{n+1}}{n^s\ln n}=0 ?$$ Since the ...
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Integration With i

Why does this approach to integration not work? If there is an integral $1/\sqrt{a^2-x^2}$, the answer is $\arcsin(x/a)$. But if the integral is $1/\sqrt{x^2-a^2}$ then it is $\log(x+\sqrt{x^2-a^2})$. ...
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Contour integration of complex number confuses me, still.

Given $f(z) = (x^2+y)+i(xy)$ and we integrate it using the Parabola Contour. For a parabola, $\gamma(t) = t + it^2$. So, $f(\gamma(t)) = 2t^2 + it^3$. What was ...
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Find the integral of $\overline{z}$

Question: Find $\int\overline{z}$, when the contour is a parabola. Interval is from 0 to 1. My Attempt: $z = x + iy \Rightarrow \overline{z} = x - iy$ $f(z) = x - iy$ Since the contour is a ...